72–75. {Use of Tech} Practical sequences
Consider the fo...

Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

d.Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist.

Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

Accepted Answer

Welcome back, everyone. A scientist injects 150 mg of a substance into a dust subject. Each hour, 8% of the substance is removed from the body. Lets n be the amount of substance remaining after n hours where S 0 equals 150 mg. Estimate the limit of the sequencesn or state that it does not exist. For this problem, we know that the first term is 150 mg. We know that each hour 8% are removed, right? So the amount remaining after N hours is going to be 100%, the initial amount, minus 8%, and that's 92% remaining. So we can write our next term as one, the amount remaining in 1 hour, as 0.92, which is the decimal form of 92%. Multiplied by S0 the previous amount, then S2 is going to be. 0.92 of the previous amount which is. 0.92 S 0. And it can be simplified into 0.92 squared as 0. And now let's write an additional term for clarity. So, S3 is going to be 0.92 of S2. So that's 0.92 multiplied by 0.92 squared as 0, which can be written as 0.92 cubed as 0 and so on. Notice that we have a basic geometric sequence as n is equal to. 0.92 raises the power of N because our index is equal to the exponent of the common ratio R multiplied by S0, right? And we can substitute for S 0, so that's 150 multiplied by 0.92 raise the power of N. Now that we have our general term formula, we want to estimate the limit of the sequence, meaning we're evaluating the limit as n approaches infinity of SN. That's a limit. As n approaches infinity of 150 multiplied by 0.92 raises the power of n. Using the properties of limits, we can factor out 150 because it's a constant. And multiply it by the limit as n approaches infinity of 0.9 to raise the power of n. Now, as our exponent approaches infinity, since the absolute value of 0.92, which is 0.92, is less than 1, the limit is going to be zero, right? Whenever we have an absolute value of a number that is less than 1. And our exponent approaches infinity. The limit is 0. Eventually our terms decrease and decrease, right, and they approach 0. So we're going to get 150 multiplied by 0 because. We already know that this limit is going to be 0. And multiplying 150 by 0, we're going to get 0. That's our final answer and thank you for watching.

Key Concepts

Geometric Sequences

Limit of a Sequence

Use of Technology in Calculus

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